The integral of the kinetic electrostatic surface wave dispersion relation is evaluated to determine the wave frequency and the corresponding Landau damping rate for the surface mode of dust ion-acoustic waves including the effect of ion temperature by using the general perturbation and the transverse truncation methods. It is demonstrated that the increase in ion temperature enhances the wave frequency. The effect of ion temperature is more prominent in the range of large wave numbers, and the wave frequency increases in proportion to a quarter power of the ion temperature in the realm of large wave number. The Landau damping of the dust ion-acoustic surface wave is found to be suppressed as the ion temperature increases. However, the effect of ion temperature becomes negligible as the wave number increases. The effects of dust charge and electron density on the Landau damping rate are also presented in this work.